Volume 2: The Fourier Transform.

Kameron Lightheart
MATH 321 Section 2
11/29/18

from matplotlib import pyplot as plt
from scipy.io import wavfile
import numpy as np
import IPython
from scipy import fftpack as fftpack
from scipy.fftpack import fft, ifft
from scipy.signal import fftconvolve
from scipy.fftpack import fft2, ifft2
import imageio

Part 1: The Discrete Fourier Transform

plt.rcParams["figure.dpi"] = 300             # Fix plot quality.
plt.rcParams["figure.figsize"] = (12,3)      # Change plot size / aspect (you may adjust this).

class SoundWave(object):
    """A class for working with digital audio signals."""

    # Problem 1.1
    def __init__(self, rate, samples):
        """Set the SoundWave class attributes.

        Parameters:
            rate (int): The sample rate of the sound.
            samples ((n,) ndarray): NumPy array of samples.
        """
        self.rate = rate
        self.samples = samples
        n = len(self.samples)
        indicies = np.arange(n)
        self.frequencies = indicies * self.rate / n

    # Problems 1.1 and 1.7
    def plot(self, myBool=False):
        """Plot the graph of the sound wave (time versus amplitude)."""
        n = len(self.samples)
        if myBool:
            Samples = fftpack.fft(self.samples)
            
            fig, ax1 = plt.subplots(1,2)
            # Plot the graph of the sound wave
            domain = np.linspace(0, (n/self.rate), n)
            ax1[0].plot(domain, self.samples)
            ax1[0].set_title("Soundwave Plot")
            ax1[0].set_ylim(-32768.0,32767.0)
            ax1[0].set_xlabel("Seconds")
            ax1[0].set_ylabel("Amplitude")
            
            indicies = np.arange(n)
            frequencies = indicies * self.rate / n
            self.frequencies = frequencies
            magnitude = np.abs(Samples)
            self.magnitude = magnitude
            
            ax1[1].plot(frequencies[:n//2], magnitude[:n//2])
            ax1[1].set_title("Frequencies vs. Magnitude")
            ax1[1].set_xlabel("Frequencies (Hz)")
            ax1[1].set_ylabel("Magnitude")
                           
            plt.show()
        else:
            # Plot the graph of the sound wave
            domain = np.linspace(0, (n/self.rate), n)
            plt.plot(domain, self.samples)
            plt.suptitle("Soundwave Plot")
            plt.ylim(-32768.0,32767.0)
            plt.xlabel("Seconds")
            plt.ylabel("Amplitude")
            plt.show()

    # Problem 1.2
    def export(self, filename, force=False):
        """Generate a wav file from the sample rate and samples. 
        If the array of samples is not of type np.int16, scale it before exporting.

        Parameters:
            filename (str): The name of the wav file to export the sound to.
        """
        
        self.filename = filename
        # Scale samples if not right dtype or force = false
        # Then write .wav file
        if (force or (self.samples.dtype != np.int16)):
            scaled_samples = np.int16((self.samples * 32767.0) / np.max(np.abs(self.samples)))
            wavfile.write(filename, self.rate, scaled_samples.real)
        else:
            wavfile.write(filename, self.rate, self.samples)
    
    # Problem 1.4
    def __add__(self, other):
        """Combine the samples from two SoundWave objects.

        Parameters:
            other (SoundWave): An object containing the samples to add
                to the samples contained in this object.
        
        Returns:
            (SoundWave): A new SoundWave instance with the combined samples.

        Raises:
            ValueError: if the two sample arrays are not the same length.
        """
        if len(self.samples) != len(other.samples):
            raise ValueError("Samples are not the same length!!!")
        # Add frequencies together to make chord
        new_samples = self.samples + other.samples
        return SoundWave(self.rate, new_samples)
        

    # Problem 1.4
    def __rshift__(self, other):
        """Concatentate the samples from two SoundWave objects.

        Parameters:
            other (SoundWave): An object containing the samples to concatenate
                to the samples contained in this object.

        Raises:
            ValueError: if the two sample rates are not equal.
        """
        if (self.rate != other.rate):
            raise ValueError("Rates don't match!!!")
        # Add other note to end of self note frequencies
        return SoundWave(self.rate, np.hstack((self.samples,other.samples)))

    
    # Problem 2.1
    def __mul__(self, other):
        """Convolve the samples from two SoundWave objects using circular convolution.
        
        Parameters:
            other (SoundWave): An object containing the samples to convolve
                with the samples contained in this object.
        
        Returns:
            (SoundWave): A new SoundWave instance with the convolved samples.

        Raises:
            ValueError: if the two sample rates are not equal.
        """
        A = self.samples
        B = other.samples
        if (self.rate != other.rate):
            raise ValueError("Rates don't match!!!")
        # Force A and B to be the same length by adding zeros to end
        if (len(A) != len(B)):
            if (len(A) > len(B)):
                diff = len(A) - len(B)
                extra = np.zeros(diff)
                B = np.hstack((extra, B))
            else:
                diff = len(B) - len(A)
                extra = np.zeros(diff)
                A = np.hstack((A, extra))
        # Compute DFT of A and B
        fftA = fft(A)
        fftB = fft(B)
        
        # Compute hadamard of the two
        hadamard = fftA * fftA
        
        # Return inverse DFT of this hadamard
        samples = ifft(hadamard)
        return SoundWave(self.rate,samples)
            

    # Problem 2.2
    def __pow__(self, other):
        """Convolve the samples from two SoundWave objects using linear convolution.
        
        Parameters:
            other (SoundWave): An object containing the samples to convolve
                with the samples contained in this object.
        
        Returns:
            (SoundWave): A new SoundWave instance with the convolved samples.

        Raises:
            ValueError: if the two sample rates are not equal.
        """
        A = self.samples
        B = other.samples
        if (self.rate != other.rate):
            raise ValueError("Rates don't match!!!")
        n = len(A)
        m = len(B)
        size = np.ceil(np.log2(n + m - 1))
        diffA = np.int(2**size - len(A))
        extraA = np.zeros(diffA)
        A = np.hstack((A,extraA))
        diffB = np.int(2**size - len(B))
        extraB = np.zeros(diffB)
        B = np.hstack((B,extraB))
        
        # Compute DFT of A and B
        fftA = fft(A)
        fftB = fft(B)
        
        # Compute hadamard of the two
        hadamard = fftA * fftA
        
        # Return inverse DFT of this hadamard
        samples = ifft(hadamard)
        
        return SoundWave(self.rate, samples[:n+m-1])
        

    # Problem 2.4
    def clean(self, low_freq, high_freq):
        """Remove a range of frequencies from the samples using the DFT. 

        Parameters:
            low_freq (float): Lower bound of the frequency range to zero out.
            high_freq (float): Higher boound of the frequency range to zero out.
        """
        # Compute DFT
        DFT = fft(self.samples)
        n = len(DFT)
        # Get low and high indicies to erase
        low_freq = np.int(low_freq * n/self.rate)
        high_freq = np.int(high_freq * n/self.rate)
        # Set values between low and high indicies to 0
        DFT[low_freq : high_freq] = 0
        # DFT is symmetric so set other ends low and high indicies to 0
        DFT[n - high_freq : n - low_freq] = 0
        # Inverse DFT back to original samples
        IDFT = ifft(DFT)
        self.samples = IDFT.real
        
    def play(self, filename=None):
        if filename is None:
            return IPython.display.Audio(rate=self.rate, data=self.samples)
        else:
            return IPython.display.Audio(filename)

Problem 1.1

• Implement SoundWave.__init__().
• Implement SoundWave.plot().
• Use the scipy.io.wavefile.read() and the SoundWave class to plot tada.wav.

# Read in tada.wav file
rate, samples = wavfile.read("tada.wav")
# Create SoundWave object and plot
SW = SoundWave(rate, samples) 
SW.plot()

Problem 1.2

• Implement SoundWave.export().
• Use the export() method to create two new files containing the same sound as tada.wav: one without scaling, and one with scaling (use force=True).
• Use IPython.display.Audio() to embed the original and new versions of tada.wav in the notebook.

# Play tada normal
IPython.display.Audio("tada.wav")

# Play tada new file, no force
SW.export("newfile1.wav")
SW.play("newfile1.wav")

# PLay tada new file, force
SW.export("newfile3.wav", force=True)
SW.play("newfile3.wav")

Problem 1.3

• Implement generate_note().
• Use generate_note() to create an A tone that lasts for two seconds. Embed it in the notebook.

def generate_note(frequency, duration):
    """Generate an instance of the SoundWave class corresponding to 
    the desired soundwave. Uses sample rate of 44100 Hz.
    
    Parameters:
        frequency (float): The frequency of the desired sound.
        duration (float): The length of the desired sound in seconds.
    
    Returns:
        sound (SoundWave): An instance of the SoundWave class.
    """
    
    rate = 44100
    
    num_samples = duration * rate
    # Make domain
    domain = np.linspace(0, num_samples / rate, num_samples)
    # Create samples with sin function
    samples = np.sin(2 * np.pi * domain * frequency)
    
    return SoundWave(rate, samples)
    
    

#Set frequency variables
A = 440
B = 493.88
C = 523.25
D = 587.33
E = 659.25
F = 698.46
G = 783.99
A_up = 880

# Make 2 second A note
A_note = generate_note(A, 2)
# PLay note
IPython.display.Audio(rate=A_note.rate, data=A_note.samples)

Problem 1.4

• Implement SoundWave.__add__().
• Generate a three-second A minor chord (A, C, and E) and embed it in the notebook.
• Implement SoundWave.__rshift__().
• Generate the arpeggio A$\,\rightarrow\,$C$\,\rightarrow\,$E, where each tone lasts one second, and embed it in the notebook.

A_note = generate_note(A, 3)
B_note = generate_note(B, 3)
C_note = generate_note(C, 3)
E_note = generate_note(E, 3)

# Make and play chord
A_C_E = A_note + C_note + E_note
IPython.display.Audio(rate=A_C_E.rate, data=A_C_E.samples)

A_note = generate_note(A, 1)
C_note = generate_note(C, 1)
E_note = generate_note(E, 1)

# PLay A then C then E one second each
A_C_E = A_note >> C_note >> E_note
IPython.display.Audio(rate=A_C_E.rate, data=A_C_E.samples)

Problem 1.5

• Implement simple_dft() with the formula for $c_k$ given below.
• Use np.allclose() to check that simple_dft() and scipy.fftpack.fft() give the same result (after scaling).

$$ c_k = \frac{1}{n}\sum_{\ell=0}^{n-1} f_\ell e^{-2 \pi i k \ell\, /\, n} $$

def simple_dft(samples):
    """Compute the DFT of an array of samples.

    Parameters:
        samples ((n,) ndarray): an array of samples.
    
    Returns:
        ((n,) ndarray): The DFT of the given array.
    """
    n = len(samples)
    l_list = np.arange(n).reshape(n,1) 
    # Build DFT matrix
    W = np.exp((-2j * np.pi/n) * (l_list @ l_list.T))
    return W @ samples / n
    
    
    

# Make random array into sound
array = np.random.random((128,))
SWarray = SoundWave(rate, array)
# Calculate DFt, compare to Scipy dft
myDFT = simple_dft(SWarray.samples)
correctDFT = fftpack.fft(SWarray.samples)
correctDFT /= len(SWarray.samples)
# Check if close enough to correct
np.allclose(myDFT, correctDFT)

True

Problem 1.6

• Implement simple_fft().
• Generate an array of $8192$ random samples and take its DFT using simple_dft(), simple_fft(), and scipy.fftpack.fft(). Print the runtimes of each computation.
• Use np.allclose() to check that simple_fft() and scipy.fftpack.fft() give the same result (after scaling).

def simple_fft(samples, threshold=1):
    """Compute the DFT using the FFT algorithm.
    
    Parameters:
        samples ((n,) ndarray): an array of samples.
        threshold (int): when a subarray of samples has fewer
            elements than this integer, use simple_dft() to
            compute the DFT of that subarray.
    
    Returns:
        ((n,) ndarray): The DFT of the given array.
    """
    n = len(samples) # assumed to be a power of 2
    
    if n <= threshold: #this cutoff to be optimized, also a power of 2
        return simple_dft(samples)
    else:
        f_even = simple_fft(samples[::2]) # FFT of even indexed entries of f
        f_odd = simple_fft(samples[1::2]) #FFT of odd indexed entries of f
        w = np.exp((-2j * np.pi/n) * np.arange(n))
        first_sum = f_even + w[:n//2] * f_odd
        second_sum = f_even + w[n//2:] * f_odd
        return 0.5 * np.concatenate([first_sum, second_sum])

# Get random 8192x1 array, create sound object
array = np.random.random((8192,))
SWarray = SoundWave(rate, array)
# Time all three functions
%time myDFT = simple_dft(SWarray.samples)
%time myFFT = simple_fft(SWarray.samples)
%time correctDFT = fftpack.fft(SWarray.samples)
correctDFT /= len(SWarray.samples)
# Check if FFT is correct
np.allclose(myFFT, correctDFT)

Wall time: 5.09 s
Wall time: 166 ms
Wall time: 8.01 ms

True

Problem 1.7

• Modify SoundWave.plot() so that it accepts a boolean. When the boolean is True, take the DFT of the stored samples and plot (in a new subplot) the frequencies present on the $x$-axis and the magnituds of those frequences on the $y$-axis. Only the display the first half of the plot, and adjust the $x$-axis so that it correctly shows the frequencies in Hertz.
• Display the plot of the DFT of an A tone.
• Display the plot of the DFT of an A minor chord.

A_note = generate_note(A,1)
rate = 44100
SW = SoundWave(rate, A_note.samples)
# Plot A and its frequencies vs magnitude plot
SW.plot(True)

SW = SoundWave(rate, A_C_E.samples)
# Plot A minor chord and its frequencies vs magnitude plot
SW.plot(True)

rate, samples = wavfile.read("tada.wav")
SW = SoundWave(rate, samples)
# Plot tada.wav file with its frequencies vs magnitude plot
SW.plot(True)

# Play mystery chord
rate, samples = wavfile.read("mystery_chord.wav")
mystery = SoundWave(rate, samples)
IPython.display.Audio(rate=mystery.rate, data=mystery.samples)

Problem 1.8

Use the DFT to determine the individual notes that are present in mystery_chord.wav.

# Read mystery file
rate, samples = wavfile.read("mystery_chord.wav")
mystery = SoundWave(rate, samples)
# Plot mystery chord
mystery.plot(True)
# Get mystery chord's magnitudes and indicies
magnitude = mystery.magnitude
frequency = mystery.frequencies
n = len(mystery.samples)
# Sort magnitudes
sort = np.argsort(magnitude[:n//2])[::-1]
# Get top notes
note1 = frequency[sort[0]]
note2 = frequency[sort[1]]
note3 = frequency[sort[2]]
note4 = frequency[sort[3]]
print(note1.real)
print(note2.real)
print(note3.real)
print(note4.real)
#note1 = generate_note(note1.real, 1)
#IPython.display.Audio(note1.rate, note1.samples)

A_note = generate_note(A, 4)
G_note = generate_note(G, 4)
C_note = generate_note(C, 4)
D_note = generate_note(D, 4)

note = A_note + G_note + C_note + D_note 
IPython.display.Audio(rate=note.rate, data=note.samples)

440.0
784.0
523.25
587.5

The notes are...A C D G !!!

Part 2: Convolution and Filtering.

Problem 2.1

• Implement SoundWave.__mul__() for circular convolution.
• Generate 2 seconds of white noise at the same sample rate as tada.wav.
• Compute the circular convolution of tada.wav and the white noise. Embed the result in the notebook.
• Append the circular convolution to itself and embed the result in the notebook.

rate, samples = wavfile.read("tada.wav")
tada = SoundWave(rate, samples)
rate = 22050
white_noise = np.random.randint(-32767, 32767, rate*2, dtype=np.int16)
white_noise = SoundWave(rate, white_noise)

convolved = tada * white_noise
convolved_appended = convolved >> convolved
IPython.display.Audio(rate=convolved_appended.rate, data=convolved_appended.samples)

C:\Users\kamer\Anaconda3\lib\site-packages\IPython\lib\display.py:124: ComplexWarning: Casting complex values to real discards the imaginary part
  data = np.array(data, dtype=float)

Problem 2.2

• Implement SoundWave.__pow__() for linear convolution.
• Time the linear convolution of CGC.wav and GCG.wav using SoundWave.__pow__() and scipy.signal.fftconvolve().
• Embed the two original sounds and their convolutions in the notebook. Check that the convolutions with SoundWave.__pow__() and scipy.signal.fftconvolve() sound the same.

rateCGC, sampsCGC = wavfile.read("CGC.wav")
rateGCG, sampsGCG = wavfile.read("GCG.wav")
CGC = SoundWave(rateCGC, sampsCGC)
GCG = SoundWave(rateGCG, sampsGCG)

IPython.display.Audio(rate=CGC.rate, data=CGC.samples)

IPython.display.Audio(rate=GCG.rate, data=GCG.samples)

# Time both my convolve and scipy's, play convolved using __pow__ method
%time CGCconv = CGC**GCG
%time CGCconvSci = fftconvolve(CGC,GCG)
IPython.display.Audio(rate=CGCconv.rate, data=CGCconv.samples)

Wall time: 335 ms
Wall time: 65.5 ms

Problem 2.3

Use SoundWave.__pow__() or scipy.signal.fftconvolve() to compute the linear convolution of chopin.wav and balloon.wav. Embed the two original sounds and their convolution in the notebook.

rate, samples = wavfile.read("chopin.wav")
chopin = SoundWave(rate, samples)
rate, samples = wavfile.read("balloon.wav")
balloon = SoundWave(rate,samples)
conv = chopin**balloon
IPython.display.Audio(rate=rate, data=conv.samples)

Problem 2.4

• Implement SoundWave.clean().
• Clean noisy1.wav by filtering out frequencies from $1250$-$2600$ Hz. Embed the original and the cleaned versions in the notebook.
• Clean noisy2.wav. Embed the original and the cleaned versions in the notebook.

rate, samples = wavfile.read("noisy1.wav")
noisy1 = SoundWave(rate, samples)
noisy1.clean(1250,2600)
noisy1.plot(True)
IPython.display.Audio(rate=noisyClean.rate, data=noisy1.samples)

rate, samples = wavfile.read("noisy2.wav")
noisy2 = SoundWave(rate, samples)
# Plot 
noisy2.plot(True)
# Clean noise
noisy2.clean(1200, 15000)
# Plot cleaned graph
noisy2.plot(True)
# Play audio
IPython.display.Audio(rate=noisyClean.rate, data=noisy2.samples)

Problem 2.5

• Clean vuvuzela.wav by filtering bad frequencies out of the left and right channels individually.
• Recombine the left and right channels and embed the result in the notebook.

rate, samples = wavfile.read("vuvuzela.wav")
vuv1 = SoundWave(rate, samples[:,0])
vuv2 = SoundWave(rate, samples[:,1])
# Clean noise
vuv1.clean(200, 500)
vuv2.clean(200, 500)
# Combine the two clean samples
vuv = SoundWave(vuv1.rate, np.vstack((vuv1.samples,vuv2.samples)))
# Play audio
IPython.display.Audio(rate=vuv.rate, data=vuv.samples)

Problem 2.6

• Clean up license_plate.png so that the year printed on the sticker in the bottom right corner of the plate is legible.
• Display the original and cleaned images.

# read in image
im = imageio.imread("license_plate.png")
plt.imshow(im, cmap="gray")
plt.show()

 # Compute DFT
im_dft = fft2(im)

# Fix brightness
im_dft[32:38,98:110] = 99
im_dft[178:184,328:336] = 99
im_dft[110:120,125:140] = 99
im_dft[65:75,195:210] = 99
im_dft[140:155,225:240] = 99
im_dft[100:110,300:310] = 99

# Inverse back to original, now fixed
im_idft = np.real(ifft2(im_dft))
plt.imshow(im_idft, cmap="gray")
plt.show()

Clipping input data to the valid range for imshow with RGB data ([0..1] for floats or [0..255] for integers).

The year on the sticker is...13